Optimal vaccine for human papillomavirus and age-difference between partners

We introduce a two sex age-structured mathematical model to describe the dynamics of HPV disease with childhood and catch up vaccines. We find the basic reproduction number (R₀) for the model and show that the disease free equilibrium is locally asymptotically stable when R₀≤1. We introduce an optimal control problem and prove that optimal vaccine solution exists and is unique. Using numerical simulation, we show that 77% childhood vaccination controls HPV disease in a 20 years period, but 77% catch up vaccine does not. In fact, catch up vaccine has a slight effect on HPV disease when applied alone or with childhood vaccine. We estimate the optimal vaccine needed to control HPV in a 25 year period. We show that reducing the partners between youths and adults is an effective way in reducing the number of HPV cases, the vaccine needed and the cost of HPV. In sum, we show that choosing partners within the same age group is more effective in controlling HPV disease than providing adult catch up vaccination.